Select Connected Bones in Pose Mode
It is difficult to diagnose this without the Blender file. One issue could be that although the bones are parented to one another, they are not "connected". To correct this, go into edit mode, and make sure that "connected" is enabled for the bones:


Using L will select all bones that are connected though, which may include the entire rig, not just the hand, depending on your setup.

Alternatively, you should consider using Bone Groups for making such selections: https://docs.blender.org/manual/en/dev/rigging/armatures/properties/bone_groups.html